Given any \(V^*=V\in L_{p+\gamma }(n_\theta )\) , set \(\dot{H}_V=\dot{\Delta }^{\frac{n}{2p}}+\dot{\lambda }(V)\) . We have the usual formula (see, e.g.,
[1]}, [2]}),
\(\sum _{j\ge 0} \left|\lambda ^{-}_j\big (\dot{H}_V\big )\right|^\gamma = \int _{-\infty }^0 (-t)^\gamma dN^{-}\big (\dot{H}_V;t\big )= \gamma \int _0^\infty t^{\gamma -1}N^{-}\big (\dot{H}_V;-t\big )dt.\) 
By the CLR inequality (REF ) we have
\(N^{-}\big (\dot{H}_V;-t\big ) = N^{-}\big (\dot{H}_{V+t}\big ) \le c_+(2p)^{2p}\nu _0(n)\tau \big [(V+t)_{-}^p\big ].\) 
Thus,
\(\sum _j \left|\lambda ^{-}_j\big (\dot{H}_V\big )\right|^\gamma \le \gamma c_+(2p)^{2p}\nu _0(n) \int _0^\infty t^{\gamma -1}\tau \big [|(V+t)_{-}|^p\big ]dt.\) 
Let \(E_v=\mathbb {1}_{(-\infty ,v]}(-V)\) , \(v\in \mathbb {R}\) , be the spectral measure of \(-V\) . The spectral measure of \((V+t)_{-}=(-V-t)_+\) , \(t\ge 0\) , is
\(\mathbb {1}_{[0,\infty )}(v)E_{v+t}\) , and hence
\(\tau \big [(V+t)_{-}^p\big ]= \int _0^\infty v^p d(\tau _*E_{v+t}) = \int _{t}^\infty (v-t)^p d(\tau _*E_v).\) 
Therefore, we have
\(\int _0^\infty t^{\gamma -1}\tau \big [(V+t)_{-}^p\big ] dt& = \int _0^\infty \bigg (\int _{t}^\infty t^{\gamma -1} (v-t)^p d(\tau _*E_v)\bigg ) dt\\& = \int _0^\infty \bigg (\int _{0}^\lambda t^{\gamma -1} (v-t)^p dt\bigg ) d(\tau _*E_v).\) 

Note that
\(\int _{0}^\lambda t^{\gamma -1} (v-t)^p dt =v^{p+\gamma } \int _0^1t^\gamma (1-t)^{p}dt=v^{p+\gamma } B(\gamma ,p+1),\) 
where \(B(x,y)=\Gamma (x+y)^{-1}\Gamma (x)\Gamma (y)\)  is the beta function. Thus,
\(\int _0^\infty t^{\gamma -1}\tau \big [(V+t)_{-}^p\big ] dt = B(\gamma ,p+1)\int _0^\infty v^{p+\gamma } d(\tau _*E_v)=B(\gamma ,p+1)\tau \big [|V_{-}|^{p+\gamma }\big ].\) 
Combining this with (REF ) gives the result.
